Let $k$ be an algebraically closed field, $A$ a finite dimensional (unital associative) $k$-algebra of finite dimension, and $G$ a torus over $k$ acting on $A$.
What does $k$-algebra $A \rtimes G$ stand for? More specifically how to define addition, multiplication, and scaler multiplication?
The only usage of $\rtimes$ that I know is that of semidirect product and trivial extension algebra. But none of them make sense here. I came across this in the proposition 5.1 in this paper (alternative link); the author says the statement is classic and doesn't explain what he meant by the symbol. Could you help me?